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Si se sabe que los vertices de un triangulo son los puntos \( A(2,-1,6), B(-4,3,0) \) y \( C(-2,-3,4) \), Cual es la forma vectorial de la recta que contiene la mediana trazada desde el vertice A a. \( L: \quad(x, y, z)=(2,-1,6)+\lambda(-1,-1,4) \) b. \( L: \quad(x, y, z)=(2,-1,6)+\lambda(-5,1,-4) \) c. \( L: \quad(x, y, z)=(2,-1,6)+\lambda(5,-1,4) \) d. Ninguna de las mencionadas

Ask by Rose Cummings. in Colombia
Dec 17,2024

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b. \( L: \quad(x, y, z)=(2,-1,6)+\lambda(-5,1,-4) \)

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Para resolver esto, primero encontramos el punto medio del lado \( BC \). El punto medio \( M \) se calcula como: \[ M = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, \frac{z_B + z_C}{2} \right) = \left( \frac{-4 - 2}{2}, \frac{3 - 3}{2}, \frac{0 + 4}{2} \right) = \left( -3, 0, 2 \right) \] Luego, la dirección de la mediana desde \( A \) a \( M \) es el vector \( \overrightarrow{AM} \): \[ \overrightarrow{AM} = M - A = (-3 - 2, 0 + 1, 2 - 6) = (-5, 1, -4) \] La forma vectorial de la recta que contiene la mediana es: \[ L: \ (x, y, z) = (2, -1, 6) + \lambda(-5, 1, -4) \] Por lo tanto, la respuesta correcta es la opción **b**.

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